Computational Game Theory Amos Fiat Spring 2012

Computational Game TheoryAmos FiatSpring 2012 Social Welfare, Arrow + Gibbard-Satterthwaite, VCG+CPP Adapted from slides by Uri FeigeRobiKrauthgamerMoniNaor TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA

Socialchoice or Preference Aggregation • Collectively choose among outcomes • Elections, • Choice of Restaurant • Rating of movies • Who is assigned what job • Goods allocation • Should we build a bridge? • Participants have preferences over outcomes • A social choice function aggregates those preferences and picks an outcome

Voting If there are two options and an odd number of voters • Each having a clear preference between the options Natural choice: majority voting • Sincere/Truthful • Monotone • Merging two sets where the majorities are the same preserves majority • Order of queries has no significance

When there are more than two options: a10, a1, … , a8 If we start pairing the alternatives: • Order may matter Assumption: n voters give their complete ranking on set A of alternatives • L – the set of linear orderson A (permutations). • Each voter i provides Ái2 L • Input to the aggregator/voting rule is (Á1,Á2,… ,Án) Goals A function f: Ln A is called a social choice function • Aggregates voters preferences and selects a winner A function W: Ln L is called a social welfare function • Aggergates voters preference into a common order am a2 a1 A

Examples of voting rules • Scoring rules: defined by a vector (a1, a2, …, am) • Being ranked ith in a vote gives the candidate ai points • Plurality: defined by (1, 0, 0, …, 0) • Winner is candidate that is ranked first most often • Veto: is defined by (1, 1, …, 1, 0) • Winner is candidate that is ranked last the least often • Borda: defined by (m-1, m-2, …, 0) • Plurality with (2-candidate) runoff: top two candidates in terms of plurality score proceed to runoff. • Single Transferable Vote (STV, aka. Instant Runoff): candidate with lowest plurality score drops out; for voters who voted for that candidate: the vote is transferred to the next (live) candidate • Repeat until only one candidate remains Jean-Charles de Borda 1770

Marquis de Condorcet • There is something wrong with Borda! [1785] Marie Jean Antoine Nicolas de Caritat, marquis de Condorcet 1743-1794

Condorcet criterion • A candidate is the Condorcet winner if it wins all of its pairwise elections • Does not always exist… • Condorcet paradox: there can be cycles • Three voters and candidates: • a > b > c, b > c > a, c > a > b • a defeats b, b defeats c, c defeats a • Many rules do not satisfy the criterion • For instance: plurality: • b > a > c > d • c > a > b > d • d > a > b > c • a is the Condorcet winner, but not the plurality winner • Candidates a and b: • Comparing how often a is ranked above b, to how often b is ranked above a • Also Borda: • a > b > c > d > e • a > b > c > d > e • c > b > d > e > a

Even more voting rules… • Kemeny: • Consider all pairwise comparisons. • Graph representation: edge from winner to loser • Create an overall ranking of the candidates that has as few disagreements as possible with the pairwise comparisons. • Delete as few edges as possible so as to make the directed comparison graph acyclic • Approval [not a ranking-based rule]: every voter labels each candidate as approved or disapproved. Candidate with the most approvals wins • How do we choose one rule from all of these rules? • What is the “perfect” rule? • We list some natural criteria • Honor societies • General Secretary of the UN

Arrow’s Impossibility Theorem Skip to the 20th Centrury Kenneth Arrow, an economist. In his PhD thesis, 1950, he: • Listed desirable properties of voting scheme • Showed that no rule can satisfy all of them. Properties • Unanimity • Independence of irrelevant alternatives • Not Dictatorial Kenneth Arrow 1921-

Independence of irrelevant alternatives • Independence of irrelevant alternatives: if • the rule ranks a above b for the current votes, • we then change the votes but do not change which is ahead between a and b in each vote • then a should still be ranked ahead of b. • None of our rules satisfy this property • Should they? b a a ¼ a a b a a b b b b

Arrow’s Impossibility Theorem • Every Social Welfare FunctionW over a set A of at least 3 candidates: • If it satisfies • Independence of irrelevant alternatives • Pareto efficiency: • If for all iaÁib • then aÁb where W(Á1,Á2,… ,Án) = Á • Then it is dictatorial : for all such W there exists an index i such that for all Á1,Á2,… ,Án2 L, W(Á1,Á2,… ,Án) = Ái

Proof of Arrow’s Theorem Claim: Let W be as above, and let • Á1,Á2,…,Ánand Á’1,Á’2,…,Á’nbe two profiles s.t. • Á=W(Á1,Á2,…,Án) and Á’=W(Á’1,Á’2,…,Á’n) • and where for all i aÁib cÁ’id Then aÁb cÁ’d Proof: suppose aÁb and c b Create a single preferenceifrom Ái and Á’i: where c is just below a and d just above b. LetÁ=W(Á1,Á2,…,Án) We must have: (i)aÁb (ii) c Á a and (iii) b Á d And therefore c Á d and c Á’d

Proof of Arrow’s Theorem: Find the Dictator Claim: For arbitrary a,b2 A consider profiles Voters Hybrid argument 1 • Change must happen at some profile i* • Where voter i* changed his opinion 2 … n Claim: this i* is the dictator! 0 1 2 n aÁb bÁa Profiles

Proof of Arrow’s Theorem: i* is the dictator Claim: for any Á1,Á2,…,Án and Á=W(Á1,Á2,…,Án) and c,d 2 A. If c Ái* d then c Á d. Proof: take e c,d and • for i<i* move e to the bottom of Ái • for i>i* move e to the top of Ái • for i* put e between c and d For resulting preferences: • Preferences of e and c like a and b in profile i*. • Preferences of e and d like a and b in profile i*-1. c Á e e Á d Thereforec Á d

Social welfare vs. Social Choice A function f: Ln A is called a social choice function • Aggregates voters preferences and selects a winner A function W: Ln L is called a social welfare function • Aggergates voters preference into a common order • We’ve seen: • Arrows Theorem: Limitations on Social Welfare functions • Next: • Gibbard-Satterthwaite Theorem: Limitations on Incentive Compatible Social Choice functions

Strategic Manipulations • A social choice functionf can be manipulated by voter i if for some Á1,Á2,…,Ánand Á’iand we have a=f(Á1,…Ái,…,Án) anda’=f(Á1,…,Á’i,…,Án) but aÁia’ voter i prefers a’ over a and can get it by changing her vote from her true preference Áito Á’i f is called incentive compatible if it cannot be manipulated

Gibbard-Satterthwaite Impossibility Theorem • Suppose there are at least 3 alternatives • There exists no social choice functionf that is simultaneously: • Onto • for every candidate, there are some preferences so that the candidate alternative is chosen • Nondictatorial • Incentive compatible

Proof of the Gibbard-Satterthwaite Theorem Construct a Social Welfare function Wf (total order) based on f. Wf(Á1,…,Án)=Á where aÁbiff f(Á1{a,b},…,Án{a,b})=b Keep everything in order but move a and b to top

Homework • Complete in full the proof of the Gibbard-Satterthwaite Theorem

Social choice in the quasi linear setting • Set of alternatives A • Who wins the auction • Which path is chosen • Who is matched to whom • Each participant: a type function ti:AR • Note: real value, not only a Áib • Participant = agent/bidder/player/etc.

Mechanism Design • We want to implement a social choice function • (a function of the agent types) • Need to know agents’ types • Why should they reveal them? • Idea: Compute alternative (a in A) and payment vector p • Utility to agent i of alternative a with payment pi is ti(a)-pi Quasi linear preferences

The setting • A social planner wants to choose an alternative according to players’ types: f : T1 × ... × Tn→ A • Problem: the planner does not know the types.

Example: Vickrey’s Second Price Auction • Single item for sale • Each player has scalar value zi – value of getting item • If he wins item and has to pay p: utility zi-p • If someone else wins item: utility 0 Second price auction: Winner is the one with the highest declared valuezi. Pays the second highest bid p*=maxj izj Theorem (Vickrey): for any every z1, z2,…,zn and every zi’. Let ui be i’s utility if he bids zi and u’i if he bids zi’. Then ui¸u’i..

Direct Revelation Mechanism A direct revelation mechanism is a social choice function f: T1 T2 … Tn A and payment functions pi: T1 T2 … Tn R • Participant i pays pi(t1, t2, … tn) A mechanism (f,p1, p2,… pn) is incentive compatible in dominant strategies if for every t=(t1, t2, …,tn), i and ti’ 2 Ti: if a = f(ti,t-i) and a’ = f(t’i,t-i) then ti(a)-pi(ti,t-i)¸ti(a’)-pi(t’i,t-i) t=(t1, t2,… tn) t-i=(t1, t2,… ti-1 ,ti+1,… tn)

Vickrey Clarke Grove Mechanism A mechanism (f,p1, p2,… pn ) is called Vickrey-Clarke-Grove (VCG) if • f(t1, t2, … tn)maximizes iti(a) over A • Maximizes welfare • There are functions h1, h2,… hn where hi: T1 T2 …Ti-1Ti+1 …Tn R we have that: pi(t1, t2, … tn) = hi(t-i) - j  itj(f(t1, t2,… tn)) Does not depend on ti t=(t1, t2,… tn) t-i=(t1, t2,… ti-1 ,ti+1,… tn)

Example: Second Price Auction Recall: f assigns the item to one participant and ti(j) = 0 if j i and ti(i)=zi • f(t1, t2, … tn) = is.t.zi =maxj(z1, z2,… zn) • hi(t-i) = maxj(z1, z2, … zi-1, zi+1 ,…, zn) • pi(t) = hi(v-i) - j  itj(f(t1, t2,… tn)) If i is the winner pi(t) = hi(t-i) = maxj izj and for j i pj(t)= zi – zi = 0 A={i wins|I 2 I}

VCG is Incentive Compatible Theorem: Every VCG Mechanism (f,p1, p2,… pn) is incentive compatible Proof: Fix i, t-i, tiand t’i. Let a=f(ti,t-i) and a’=f(t’i,t-i). Have to show ti(a)-pi(ti,t-i)¸ti (a’)-pi(t’i,t-i) Utility of i when declaring ti: ti(a) + j  itj(a) - hi(t-i) Utility of i when declaring t’i: ti(a’)+ j  itj(a’)- hi(t-i) Since amaximizessocial welfare ti(a) + j  itj(a) ¸ti(a’) + j  itj(a’)

Clarke Pivot Rule What is the “right”: h? Individually rational: participants always get non negative utility ti(f(t1, t2,… tn)) - pi(t1, t2,… tn) ¸ 0 No positive transfers: no participant is ever paid money pi(t1, t2,… tn) ¸ 0 Clark Pivot rule: Choosing hi(t-i) = maxb2 Aj  itj(b) Payment of iwhen a=f(t1, t2,…, tn): pi(t1, t2,… tn) = maxb2 Aj  itj(b) -j  itj(a) i pays an amount corresponding to the total “damage” he causes other players: difference in social welfare caused by his participation

Rationality of Clarke Pivot Rule Theorem: Every VCG Mechanism with Clarke pivot payments makes no positive Payments. If ti(a) ¸0 then it is Inditidually rational Proof: Let a=f(t1, t2,… tn) maximizessocial welfare Let b 2 A maximizej  itj(b) Utility of i: ti(a) + j  itj(a) - j  itj(b) ¸j tj(a) - j tj(b) ¸ 0 Payment of i: j  itj(b) -j  itj(a) ¸ 0 from choice of b maximizes iti(a) over A

Computational Game Theory Amos Fiat Spring 2012